Nuprl Lemma : cong-tri-implies-cong-angle
∀e:BasicGeometry. ∀a,b,c,x,y,z:Point. (Cong3(abc,xyz) ⇒ a ≠ b ⇒ b ≠ c ⇒ x ≠ y ⇒ y ≠ z ⇒ abc ≅a xyz)
Proof
Definitions occuring in Statement :
geo-cong-tri: Cong3(abc,a'b'c'),
geo-cong-angle: abc ≅a xyz,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
geo-cong-angle: abc ≅a xyz,
and: P ∧ Q,
geo-cong-tri: Cong3(abc,a'b'c'),
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
basic-geometry: BasicGeometry,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T}
Lemmas referenced :
geo-between-trivial,
geo-congruent-iff-length,
geo-length-flip,
geo-between_wf,
geo-congruent_wf,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-cong-tri_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation_alt,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productIsType,
universeIsType,
applyEquality,
instantiate,
inhabitedIsType
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,x,y,z:Point.
(Cong3(abc,xyz) {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} b \mneq{} c {}\mRightarrow{} x \mneq{} y {}\mRightarrow{} y \mneq{} z {}\mRightarrow{} abc \mcong{}\msuba{} xyz)
Date html generated:
2019_10_16-PM-01_22_42
Last ObjectModification:
2018_11_07-PM-00_53_15
Theory : euclidean!plane!geometry
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